A Riemann-Roch formula for singular reductions by circle actions

TL;DR

This paper derives a Riemann-Roch formula for the invariant number of a Hamiltonian circle action on a symplectic manifold, including cases with singularities, using advanced geometric and analytical techniques.

Contribution

It introduces a new explicit local invariant for singularities and extends the Riemann-Roch formula to singular reductions with a stationary phase expansion approach.

Findings

Provides a formula involving the geometry of symplectic quotients with singularities.

Introduces a new local invariant of singularities.

Employs a complete stationary phase expansion of the Witten integral.

Abstract

We compute a Riemann-Roch formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,\omega,\mathcal{J})$ in terms of the geometry of its symplectic quotient, allowing $0$ to be a singular value of the moment map $\mathcal{J}:M\to\mathbb{R}$. The formula involves a new explicit local invariant of the singularities. Our approach relies on a complete singular stationary phase expansion of the associated Witten integral.